Why does this paper say that the Nash-equilibrium of GAN is given by a discriminator which is 0 everywhere on the data distribution?
What this paper is not saying is that the discriminator, $D_{\phi}$, always returns a scalar value of zero. What they are saying is that the generator, $G_{\theta}(z)$, has accurately learned the distribution of the input data, and that the discriminator produces the correct answer for each input from the generator. It's a mathematical description of the global optimum for a GAN. This Medium article by Jonathan Hui talks more about Nash equilibria, the Kullback-Leibler, and more.